A TrotterSuzuki approximation for Lie groups with applications to Hamiltonian simulation
Abstract
In this paper, we present a product formula to approximate the exponential of a skewHermitian operator that is a sum of generators of a Lie algebra. The number of terms in the product depends on the structure factors. When the generators have large norm with respect to the dimension of the Lie algebra, or when the norm of the effective operator resulting from nested commutators is less than the product of the norms, the number of terms in the product is significantly less than that obtained from wellknown results. We apply our results to construct product formulas useful for the quantum simulation of some continuousvariable and bosonic physical systems, including systems whose potential is not quadratic. For many of these systems, we show that the number of terms in the product can be sublinear or even subpolynomial in the dimension of the relevant local Hilbert spaces, where such a dimension is usually determined by the energy scale of the problem. Our results emphasize the power of quantum computers for the simulation of various quantum systems.
 Authors:

 Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Theoretical Division
 Publication Date:
 Research Org.:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org.:
 USDOE Laboratory Directed Research and Development (LDRD) Program
 OSTI Identifier:
 1337097
 Alternate Identifier(s):
 OSTI ID: 1421146
 Report Number(s):
 LAUR1529416
Journal ID: ISSN 00222488
 Grant/Contract Number:
 AC5206NA25396
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Journal of Mathematical Physics
 Additional Journal Information:
 Journal Volume: 57; Journal Issue: 6; Journal ID: ISSN 00222488
 Publisher:
 American Institute of Physics (AIP)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICS AND COMPUTING; Quantum Computing; Quantum Algorithms
Citation Formats
Somma, Rolando D. A TrotterSuzuki approximation for Lie groups with applications to Hamiltonian simulation. United States: N. p., 2016.
Web. https://doi.org/10.1063/1.4952761.
Somma, Rolando D. A TrotterSuzuki approximation for Lie groups with applications to Hamiltonian simulation. United States. https://doi.org/10.1063/1.4952761
Somma, Rolando D. Wed .
"A TrotterSuzuki approximation for Lie groups with applications to Hamiltonian simulation". United States. https://doi.org/10.1063/1.4952761. https://www.osti.gov/servlets/purl/1337097.
@article{osti_1337097,
title = {A TrotterSuzuki approximation for Lie groups with applications to Hamiltonian simulation},
author = {Somma, Rolando D.},
abstractNote = {In this paper, we present a product formula to approximate the exponential of a skewHermitian operator that is a sum of generators of a Lie algebra. The number of terms in the product depends on the structure factors. When the generators have large norm with respect to the dimension of the Lie algebra, or when the norm of the effective operator resulting from nested commutators is less than the product of the norms, the number of terms in the product is significantly less than that obtained from wellknown results. We apply our results to construct product formulas useful for the quantum simulation of some continuousvariable and bosonic physical systems, including systems whose potential is not quadratic. For many of these systems, we show that the number of terms in the product can be sublinear or even subpolynomial in the dimension of the relevant local Hilbert spaces, where such a dimension is usually determined by the energy scale of the problem. Our results emphasize the power of quantum computers for the simulation of various quantum systems.},
doi = {10.1063/1.4952761},
journal = {Journal of Mathematical Physics},
number = 6,
volume = 57,
place = {United States},
year = {2016},
month = {6}
}
Web of Science
Works referenced in this record:
Fractal decomposition of exponential operators with applications to manybody theories and Monte Carlo simulations
journal, June 1990
 Suzuki, Masuo
 Physics Letters A, Vol. 146, Issue 6
Quantum algorithms for fermionic simulations
journal, July 2001
 Ortiz, G.; Gubernatis, J. E.; Knill, E.
 Physical Review A, Vol. 64, Issue 2
Simulated Quantum Computation of Molecular Energies
journal, September 2005
 AspuruGuzik, A.
 Science, Vol. 309, Issue 5741
Simulating Hamiltonian Dynamics with a Truncated Taylor Series
journal, March 2015
 Berry, Dominic W.; Childs, Andrew M.; Cleve, Richard
 Physical Review Letters, Vol. 114, Issue 9
Gatecount estimates for performing quantum chemistry on small quantum computers
journal, August 2014
 Wecker, Dave; Bauer, Bela; Clark, Bryan K.
 Physical Review A, Vol. 90, Issue 2
Exponentially more precise quantum simulation of fermions in second quantization
journal, March 2016
 Babbush, Ryan; Berry, Dominic W.; Kivlichan, Ian D.
 New Journal of Physics, Vol. 18, Issue 3
Adiabatic quantum state generation and statistical zero knowledge
conference, January 2003
 Aharonov, Dorit; TaShma, Amnon
 Proceedings of the thirtyfifth ACM symposium on Theory of computing  STOC '03
Simulating physics with computers
journal, June 1982
 Feynman, Richard P.
 International Journal of Theoretical Physics, Vol. 21, Issue 67
Simulating physical phenomena by quantum networks
journal, April 2002
 Somma, R.; Ortiz, G.; Gubernatis, J. E.
 Physical Review A, Vol. 65, Issue 4
Canonical representations of sp(2 n , R )
journal, April 1992
 Rangarajan, Govindan; Neri, Filippo
 Journal of Mathematical Physics, Vol. 33, Issue 4
Simulating Chemistry Using Quantum Computers
journal, May 2011
 Kassal, Ivan; Whitfield, James D.; PerdomoOrtiz, Alejandro
 Annual Review of Physical Chemistry, Vol. 62, Issue 1
General theory of fractal path integrals with applications to many‐body theories and statistical physics
journal, February 1991
 Suzuki, Masuo
 Journal of Mathematical Physics, Vol. 32, Issue 2
Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation
journal, August 1984
 Taha, Thiab R.; Ablowitz, Mark I.
 Journal of Computational Physics, Vol. 55, Issue 2
Optimal Trotterization in universal quantum simulators under faulty control
journal, May 2015
 Knee, George C.; Munro, William J.
 Physical Review A, Vol. 91, Issue 5
Exponential improvement in precision for simulating sparse Hamiltonians
conference, January 2014
 Berry, Dominic W.; Childs, Andrew M.; Cleve, Richard
 Proceedings of the 46th Annual ACM Symposium on Theory of Computing  STOC '14
Efficient Quantum Algorithms for Simulating Sparse Hamiltonians
journal, December 2006
 Berry, Dominic W.; Ahokas, Graeme; Cleve, Richard
 Communications in Mathematical Physics, Vol. 270, Issue 2
Higher order decompositions of ordered operator exponentials
journal, January 2010
 Wiebe, Nathan; Berry, Dominic; Høyer, Peter
 Journal of Physics A: Mathematical and Theoretical, Vol. 43, Issue 6
Chemical basis of TrotterSuzuki errors in quantum chemistry simulation
journal, February 2015
 Babbush, Ryan; McClean, Jarrod; Wecker, Dave
 Physical Review A, Vol. 91, Issue 2
Simulating physics with computers
journal, June 1982
 Feynman, Richard P.
 International Journal of Theoretical Physics, Vol. 21, Issue 67
Fractal decomposition of exponential operators with applications to manybody theories and Monte Carlo simulations
journal, June 1990
 Suzuki, Masuo
 Physics Letters A, Vol. 146, Issue 6
Simulating Hamiltonian Dynamics with a Truncated Taylor Series
journal, March 2015
 Berry, Dominic W.; Childs, Andrew M.; Cleve, Richard
 Physical Review Letters, Vol. 114, Issue 9
Quantum algorithms for fermionic simulations
journal, July 2001
 Ortiz, G.; Gubernatis, J. E.; Knill, E.
 Physical Review A, Vol. 64, Issue 2
Simulating physical phenomena by quantum networks
journal, April 2002
 Somma, R.; Ortiz, G.; Gubernatis, J. E.
 Physical Review A, Vol. 65, Issue 4
Gatecount estimates for performing quantum chemistry on small quantum computers
journal, August 2014
 Wecker, Dave; Bauer, Bela; Clark, Bryan K.
 Physical Review A, Vol. 90, Issue 2
Optimal Trotterization in universal quantum simulators under faulty control
journal, May 2015
 Knee, George C.; Munro, William J.
 Physical Review A, Vol. 91, Issue 5
Chemical basis of TrotterSuzuki errors in quantum chemistry simulation
journal, February 2015
 Babbush, Ryan; McClean, Jarrod; Wecker, Dave
 Physical Review A, Vol. 91, Issue 2
Exponential Improvement in Precision for Simulating Sparse Hamiltonians
journal, January 2017
 Berry, Dominic W.; Childs, Andrew M.; Cleve, Richard
 Forum of Mathematics, Sigma, Vol. 5
Works referencing / citing this record:
Bounding the costs of quantum simulation of manybody physics in real space
journal, June 2017
 Kivlichan, Ian D.; Wiebe, Nathan; Babbush, Ryan
 Journal of Physics A: Mathematical and Theoretical, Vol. 50, Issue 30
Nearly Optimal Lattice Simulation by Product Formulas
journal, August 2019
 Childs, Andrew M.; Su, Yuan
 Physical Review Letters, Vol. 123, Issue 5
Hamiltonian Simulation by Qubitization
journal, July 2019
 Low, Guang Hao; Chuang, Isaac L.
 Quantum, Vol. 3
Compilation by stochastic Hamiltonian sparsification
journal, February 2020
 Ouyang, Yingkai; White, David R.; Campbell, Earl T.
 Quantum, Vol. 4